He becomes the nation's fourth leader in a little over two years, but despite division in his party, Mr Turnbull says he will lead a "very strong government".

"There's been a change of prime minister, but we are a very, very strong government, a very strong country with a great potential and we will realise that potential working very hard together," he said.

"This is a turn of events I did not expect, I have to tell you, but it's one that I'm privileged to undertake and one that I'm certainly up to."

Shortly before Mr Turnbull was sworn in, Mr Abbott spoke for the first time since his sudden ousting.

He warned that the persistent volatility in Australia's government could hurt the nation's standing on the global stage.

"Australia has a role to play in the struggles of the wider world: the cauldron of the Middle East and security in the South China Sea and elsewhere," he said.

"I fear that none of this will be helped if the leadership instability that's plagued other countries continues to taint us."

Mr Turnbull later had to meet with his coalition government to thrash out a new agreement with the National Party to replace the one that had been put in place by his predecessor.

Australian media reported that the agreement is thought to contain a series of promises to deliver more for people in rural communities, which make up majority of the Nationals' support.

Many conservative Nationals were upset at the removal of Mr Abbott, who is seen as more right of centre than his successor.

Mr Turnbull maintains some notoriety among the UK Conservative establishment for having been the lawyer who defeated a British attempt to stop the publication of the MI5 expose Spycatcher in the 1980s.

Despite being a Catholic, he is in favour of same-sex marriage and in his first speech to parliament as prime minister said Australians would vote on same-sex marriage after elections due next year.

He is also a staunch republican and led a previous failed campaign to remove the Queen as Australian head of state.